Volume of sphere

Introduction to Sphere : A tennis ball and a fully blown football  are some familiar objects which bring to our mind the concept of a sphere .A sphere is a three dimensional geometrical object  which can be defined as follow The set of all points in space which are equidistant from a fixed point , is called a sphere.

The fixed point is called the center of the sphere and the constant distance is called its radius. A line segment through the center of a sphere, and with the end points on the sphere . All diameters of a sphere  are of constant length , being equal to twice the radius of the sphere .Thus , if d is the length of a diameter of a sphere   of radius r  then d= 2r . The length of diameter is also called the diameter of sphere .The solid sphere is the region in sphere, bounded by sphere .

Also every point whose distance from the center is less than or equal to the radius is a point of the solid sphere .A sphere can also be considered as a solid obtained on rotating a circle about its diameter.

Volume of a sphere: The volume of a sphere (v) of radius r is given by v = 4/3 ∏r3 cubic units .Let us take an example how to volume of sphere.

Find the sphere volume  of radius 7 cm We know that formula for volume of a sphere of radius r is given by v = 4/3 ∏r3  cubic units here r = 7 cm  therefore v = 4/3 x 22/7 x 7 x 7 x 7 cm3 another example of volume of a sphere.

Calculate volume of a sphere whose surface area is 154 square cm. Let the radius of the sphere be r cm .then ,
Surface area = 154 cm2 => 4∏r2  = 154 => 4 x 22/7 x r2 = 154 => r2 = 154 x 7 / 4 x 22 = 49/4 => r = 7/2 cm , so let v be the volume of sphere .
We will use volume of a sphere equation

That is , v= 4/3∏r3   = 4/3 x 22/7 x 7/2 x 7/2 x 7/2 cm3 = 179.66cm3.

Let us take more example volume of a sphere. A sphere of diameter 6 cm is draped in a right circular cylinder vessel partly filled with water .The diameter of the cylindrical vessel is 12 cm. If the sphere is completely submerged in water, by how much will the level of water rise in the cylindrical vessel?

We have radius of sphere = 3cm, volume of a sphere formula = 4/3 ∏ r3  cm3 = 4/3 ∏ (3)³ cm³ = 36cm³, radius of cylindrical vessel = 6cm.
Suppose of water level rises by h cm and radius 6 cm = (∏x 62 x h) cm 3 = 36∏h cm3 , clearly volume of the water displaced by the sphere is equal to the volume of  the  sphere =36∏h = > h=1 cm , hence water level rises by 1 cm